Integrand size = 30, antiderivative size = 37 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\frac {3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}} \]
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Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {3574} \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\frac {3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}} \]
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Rule 3574
Rubi steps \begin{align*} \text {integral}& = \frac {3 i a (d \sec (e+f x))^{2/3}}{f \sqrt [3]{a+i a \tan (e+f x)}} \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\frac {3 d^2 (i+\tan (e+f x)) (a+i a \tan (e+f x))^{2/3}}{f (d \sec (e+f x))^{4/3}} \]
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\[\int \left (d \sec \left (f x +e \right )\right )^{\frac {2}{3}} \left (a +i a \tan \left (f x +e \right )\right )^{\frac {2}{3}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.49 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=-\frac {3 \cdot 2^{\frac {1}{3}} \left (\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} \left (\frac {d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{\frac {2}{3}} {\left (-i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i\right )}}{f} \]
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\[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{\frac {2}{3}} \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {2}{3}}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (29) = 58\).
Time = 0.62 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.89 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=-\frac {3 \, {\left (-i \cdot 2^{\frac {1}{3}} \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) - 2^{\frac {1}{3}} \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right )\right )} a^{\frac {2}{3}} d^{\frac {2}{3}}}{{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{6}} f} \]
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\[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\int { \left (d \sec \left (f x + e\right )\right )^{\frac {2}{3}} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {2}{3}} \,d x } \]
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Time = 5.79 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.19 \[ \int (d \sec (e+f x))^{2/3} (a+i a \tan (e+f x))^{2/3} \, dx=\frac {3\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{2/3}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+\sin \left (2\,e+2\,f\,x\right )+1{}\mathrm {i}\right )\,{\left (\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}\right )}^{2/3}}{2\,f} \]
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